An analog to Deuring's criterion for good reduction of elliptic curves

Details An analog to Deuring's criterion for good reduction of elliptic curves An analog to Deuring's criterion for good reduction of elliptic curves

2004-09-25

arxiv.org/abs/math/0409493v1

Mathematics

Algebraic Geometry

12 pages

Scientific paper

In this paper we study the reduction of $p$-cyclic covers of the $p$-adic line ramified at exactly four points. For $p=2$ these covers are elliptic curves and Deuring has given a criterion for when such a curve has good reduction. Here we consider the case of $p>2$ and completely determine the stable model of the cover. In particular we obtain a finite extension $R'$ of $R$ necessary for the stable reduction to be defined. No additional conditions are imposed on the geometry of the branch locus and thus this work can be viewed as a first step towards understanding the situation where branch points coalesce.

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